IDEMPOTENT TRIANGULAR MATRICES OVER COMMUTATIVE SEMIRINGS

Abstract

The structure of idempotent triangular matrices with elements on the main diagonal being 0 or 1 over a commutative ring has been fully described by Xin Hou (2021). These results have also been generalized by Stephen E. Wright (2022) when studying the structure of idempotent triangular matrices over the general rings. Furthermore, Stephen E. Wright (2022) has provided formulas for calculating the number of matrices of this type within finite rings. In this paper, we investigate the characteristic properties of idempotent triangular matrices over commutative semirings and describe the structure of such matrices in cases where the entries on their main diagonal are pairwise orthogonal idempotent elements. Simultaneously, we proceed to compute the number of idempotent triangular matrices with entries on the main diagonal being 0 or 1 when the corresponding semirings are commutative, additively idempotent, and have a finite number of elements.